Illustrated Physics
physics, drawn
Concept

Black hole entropy — where it appears

The entropy a black hole carries, a quarter of its horizon's area measured in Planck units, as found by Bekenstein and Hawking. It grows with area rather than volume, which is the origin of the holographic idea that a region's information is bounded by its surface.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

One mass, two entropies. The entropy of a solar mass as ordinary gas, generously counted at ten Boltzmann constants per proton, against the entropy of a solar-mass horizon. The first is 1.19·10⁵⁸ k and the second 1.05·10⁷⁷ k — a factor of 8.83·10¹⁸. This is why a horizon had to be given an entropy: without one, dropping anything at all through it destroys entropy and the second law fails.

The entropy that lives on a surface

Throw a cup of tea through a horizon and the entropy of the outside world falls. Either the second law is wrong or the horizon has an entropy of its own — and the only quantity available for it turns out to be its area, in units of a length made from gravity, quantum mechanics and the speed of light together.

thermodynamics · Entropy How well a distance can be known, by how long it is. The smallest uncertainty in a distance l, on logarithmic axes, by three rules. Solid: the clock argument, 1.89 (ℓₚ² l)^(1/3), which grows as the cube root of the distance. Dotted: the Planck length, the floor usually quoted, which does not grow at all. Dashed: the guess that Planck-scale jitters add like a random walk, √(ℓₚ l). a metre: 1.2·10⁻²³ m by the clock argument, 4·10⁻¹⁸ m as a random walk; the Earth to the Sun: 6.4·10⁻²⁰ m by the clock argument, 1.6·10⁻¹² m as a random walk; the observable universe: 9.2·10⁻¹⁵ m by the clock argument, 8.4·10⁻⁵ m as a random walk. By the clock argument the size of the observable universe is uncertain by about the size of an atomic nucleus; as a random walk, by about a tenth of a millimetre. All three rules meet at the Planck length.

The clock that cannot be both light and good

The Planck length is usually quoted as the smallest distance that means anything, as though every longer distance could be known to that precision. Time a distance with light against a clock and the floor turns out to rise with the distance. A light clock wanders while the light is away; a heavy clock wanders less but is closer to being a black hole; and the best compromise leaves a metre uncertain by 750 thousand million Planck lengths and the size of the observable universe by the size of an atomic nucleus. The same compromise, counted up, says a region can hold only as many distinguishable places as its surface has Planck areas.

astrophysics · Planck scale

Named alongside it

The objects these essays reach for when they reach for this one.

EntropyEvent horizonHawking temperatureHolographic boundHolographic principleMeasurementMicrostatesPlanck lengthPlanck unitsQuantum gravitySchwarzschild radiusThe second law

All concepts